Differential Calculi on Commutative Algebras

A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in very much the same way we are used to from the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of 1-forms. An example of such a product is provided by the Ito calculus of stochastic differentials. For the case where the algebra A is freely generated by `coordinates' x^i, i=1,...,n, we study calculi for which the differentials dx^i constitute a basis of the space of 1-forms (as a left A-module). These may be regarded as `deformations' of the ordinary differential calculus on R^n. For n < 4 a classification of all (orbits under the general linear group of) such calculi with `constant structure functions' is presented. We analyse whether these calculi are reducible (i.e., a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this article) of a one dimension lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi.Comment: 33 pages, LaTeX. Revision: A remark about a quasilattice and Penrose tiling was incorrect in the first version of the paper (p. 14

Differential Geometry of Group Lattices

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property that ad(S)S is contained in S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first order calculi extend to higher orders and then allow to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained.Comment: 51 pages, 11 figure

Bicomplexes and Integrable Models

We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are presented, including the nonlinear Schrodinger and sine-Gordon equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st

From Dumb Wireless Sensors to Smart Networks using Network Coding

The vision of wireless sensor networks is one of a smart collection of tiny, dumb devices. These motes may be individually cheap, unintelligent, imprecise, and unreliable. Yet they are able to derive strength from numbers, rendering the whole to be strong, reliable and robust. Our approach is to adopt a distributed and randomized mindset and rely on in network processing and network coding. Our general abstraction is that nodes should act only locally and independently, and the desired global behavior should arise as a collective property of the network. We summarize our work and present how these ideas can be applied for communication and storage in sensor networks.Comment: To be presented at the Inaugural Workshop of the Center for Information Theory and Its Applications, University of California - San Diego, La Jolla, CA, February 6 - 10, 200

On non commutative sinh-Gordon Equation

We give a noncommutative extension of sinh-Gordon equation. We generalize a linear system and Lax representation of the sinh-Gordon equation in noncommutative space. This generalization gives a noncommutative version of the sinh-Gordon equation with extra constraints, which can be expressed as global conserved currents.Comment: 7 Page

On the Delay of Network Coding over Line Networks

We analyze a simple network where a source and a receiver are connected by a line of erasure channels of different reliabilities. Recent prior work has shown that random linear network coding can achieve the min-cut capacity and therefore the asymptotic rate is determined by the worst link of the line network. In this paper we investigate the delay for transmitting a batch of packets, which is a function of all the erasure probabilities and the number of packets in the batch. We show a monotonicity result on the delay function and derive simple expressions which characterize the expected delay behavior of line networks. Further, we use a martingale bounded differences argument to show that the actual delay is tightly concentrated around its expectation

Noncommutative Geometry of Finite Groups

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late

Abelian Toda field theories on the noncommutative plane

Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by algebra-valued gauge potentials dependent on the fields. This condition can be expressed as noncommutative Leznov-Saveliev equations which make possible to define the noncommutative generalizations as systems of second order differential equations, with an infinite chain of conserved currents. The actions corresponding to these field theories are also provided. The special cases of GL(2) Liouville and $\widetilde{GL}(2)$ sinh/sine-Gordon are explicitly studied. It is also shown that from the noncommutative (anti-)self-dual Yang-Mills equations in four dimensions it is possible to obtain by dimensional reduction the equations of motion of the two-dimensional models constructed. This fact supports the validity of the noncommutative version of the Ward conjecture. The relation of our proposal to previous versions of some specific Toda field theories reported in the literature is presented as well.Comment: v3 30 pages, changes in the text, new sections included and references adde

Bi-differential calculi and integrable models

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.Comment: 24 pages, 2 figures, uses amssymb.sty and diagrams.sty, substantial extensions of examples (relative to first version